Monday, 25 August 2014

linear algebra - Is left inverse implying right inverse in matrix a property of structure?

If A is a square matrix and there exists a square matrix B such that AB=1, than it is known that BA=1. This property is proved with some properties from linear algebra. Although I've never seen it be proved just by structures of matrix multiplication, I couldn't find a counterexample of a set with structures of matrix multiplication but left inverse doesn't imply right inverse.



To be more specific, let X be a set and binary operation is defined on X. If is associative and X has left and right identity(which will be the same), than does AB=1 for some A,BX implies BA=1?



If not, what other properties of matrix multiplication should we add to this structure of (X,) in order to get the property?

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